The Left Handed Maxwell Systems Lecture 1 by Shantanu Das
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SAMEER
Shantanu Das
RR&PS
Reactor Control Division, B.A.R.C. Mumbai400085
[email protected]
The LHM Team
SAMEER Calcutta:
1.
Dr A. L. Das (Director SAMEER)
2.
Arijit Majumder (Scientist SAMEER PrincipalInvestigator)
3.
Paulami Sarkar (Scientist SAMEER)
4.
Amitesh Kumar (Research Scientist SAMEER)
5.
Saugata Chaterjee (Reseach Scientist SAMEER)
University of Calcutta:
Prof. Subal Kar (IRPE Calcutta Univ. CU; Consultant & Guide)
BARC:
Shantanu Das (Scientist RRPS/RCNDBARC; PrincipalCollaborator)
Funded by BRNS Department of Atomic Energy under MoU between SAMEER CUBARC for
three years “Setting up Programme on Left Handed Maxwell Systems” Rs. 110,00,000. Project MoU
signed on January, 2010.
General exposition & Utility of
The
Left Handed Maxwell Systems
Part1
Wave propagation is reversed
n = 1
n = −1
n = 1
No natural material (sofar) known to behave as Left Handed material.
The concept will fabricate and demonstrate this phenomena by artificially structured &
fabricated using natural materials FR4, RTDUROID, copper wiresrings to have artificial
material “MetaMaterials”, where the effective permittivity & permeability will behave as
‘negatives'; and the refractive index as ‘negative value’.
The idea of FLAT surface focusing.
Bending of electromagnetic energy in wrong way through LHM
Poynting Vector opposite to WavePropagation Vector with negative phase velocity in LHM
→
E
θ = θ
−
r
i
→
n = 1
−
k
i
θ = θ
n = 1
→
→
→
⎛
⎞
S = ⎜ E × H ⎟
r
i
i
⎝
⎠
→
LHMBackward wave
H
Region –II (negative phase velocity)
ε < 0, μ < 0 SPPregion
RegionI (positive phase velocity)
RHMForward wave
n = 1
ε > μ
i
θ
0, > 0
i
→
E
ω
d ω
→
v
=
; v
=
p
g
k
k
d k
→
→
→
⎛
⎞
S = ⎜ E × H ⎟
→
⎝
⎠
H
With negative unity relative permittivity & permeability the interface
impedance is just of freespace, so at the interface noreflection wave.
μ r
η = η
= η = 120π = 377 Ω
0
0
ε r
But the refractive index is negative unity. An Ideal Case
Negative Refractive Index NRI
A very preliminary explanation the detailed will be carried out later in other section
n ( ω ) =
ε ( ω ) μ ( ω )
r
r
In LHM these parameters are negative for certain frequency ε
< 0 , μ < 0
r
r
ε (ω ) < 0
( ) =
( )
jπ
ε ω
ε ω e
r
r
r
μ (ω ) < 0
( ) =
( )
jπ
μ ω
μ ω e
r
r
r
π
n ( ω ) =
ε ( ω ) μ ( ω ) . j
e
= − ε ( ω ) μ
( ω )
r
r
r
r
Limiting to EM up to 100GHz, presently. Optical magnetism for IR and beyond not considered. Negative n is illusive in
optical regime. The imaginary part of permittivity and permeability in those regions will be rather large, will manifest
majorly as ohmic, and radiation loss. Also at HF the sub nano structures becomes less than skin depth; whereas in uW
ranges metals can be regarded as PEC and skin depth is smaller than characteristic size of EM lattice.
μ 1 − 2
η
= η
∠ η
=
1 − 2
1 − 2
1 − 2
ε
though seems same across, perhaps needs to be re looked at?
1 − 2
There is definitely a different explanation with respect to impedance, perhaps wave impedance may also have sign change?
Materials
μr
ENG
DPS
εr
DNG
MNG
WaveMechanics Revisited
The Wave Equation obtained from Maxwell’s equation is
2
∇
2
E + k E and
= 0
is same for
positive or negative wavevector. Where
2
2
k
= ω μ ε The solution is
.
e jk r
E
E
−
=
0
for quasistatic case.
E
Is constant vector perpendicular to k
0
2
2
2
2
k = k + k + k
x
y
z
This is plane wave as none of its variable change in plane perpendicular to wavevector
k = β − jα
is complex wavenumber.
β is propagation coefficient,
α is attenuation.
Wave in zdirection is: − β −α
E
= E
j
z
z
e
e
i.e. wave declines exponentially in the z.
z
0
This exponential decay is expected and true in all cases of forward wave. If losses are nil
then α = 0
The wave number is
k
, then equal to propagation coefficient β
Wave impedance
μ
E
μ Free space wave
0
η =
=
π
r
impedance
η =
= η
1 2 0
0
0
ε
H
ε
0
r
Phase velocity is velocity with which single frequency wave travels is v
= ω / k = c /
μ ε
p h
r
r
But single frequency wave does not carry information, can be carrier for the information.
Velocity of a group of frequencies that do carry information is group
velocity and is
d ω
v
=
g
d k
For free isotropic space v
= c
ω =
p h
dispersion relation is
k No
c
change in velocity
or the wavenumber with the frequency. Refractive index also remains same with frequency.
n =
μ ε = 1
0
r
r
Interpretation macroscopically Left Hand cross Product
For the Left Handed Maxwell (LHM) Media the result of backward wave where
plane wave propagation is, opposite to the direction of energy flow, does not
follow from the wave Equation, which remains unchanged for DNG
⎛
⎞
⎛
⎞
2
E
2
E
∇ ⎜
⎟ +
2
2
k
⎜
⎟ = 0
k
= ω μ ε
where
⎝ H ⎠
⎝ H ⎠
But comes individual Curl equations, can be formed by LeftHand
for the LHM left handed curl equations are the following:
∇ ×
→
→
→
E
= + j ω μ H
k ×
E
= − ω μ
H
∇ × H
= − j ω ε E
→
→
→
k ×
H
= + ω ε E
Poynting vector tells direction of power flow (groupvelocity) and wave vector
tells the direction of phase velocity. Opposite means “BackwardWave”
For the RHM, wave propagation phase velocity is in direction of energy flow:
→
→
→
→
→
∇ × E = − j ω μ H
k × E
= + ω μ H
→
→
∇ ×
→
→
→
H
= j ω ε E
k × H
= − ω ε E
Traveling Waves in LHM and perfect imaging!!
1 PropagatingTraveling Wave (Bulk wave)
± jk z
For Large wave length
z
2
2
For
k =
k
− k
k < k wave is: A e
z
0
x
x
0
z = 0
z
/
d
= 2
z = 3d / 2
z = 2d
n = 1
−
n = 1
n = 1
ImagePlane
ObjectPlane
ε = 1
− ,μ = 1
−
r
r
x
z
y
Medium1 Medium2 Medium3
1.
The amplitude of propagating waves are always same as losses are neglected
2.
The phase goes forward by
k
d / 2 in medium1
z
3.
The phase goes backward by in medium2
k d
z
4.
The phase goes forward by
k
d / 2 in medium3
z
5.
The total phase change is zero, phase is restored. Transfer function is flat for all
propagating wavenumbers, or spatial spectral frequencies.
6.
The perfect EM image is thus possible (ideally).
7.
Zero EM (optical) path is only possible with negative refractive index (NRI)
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